A generalized Grothendieck inequality and entanglement in XOR games

TL;DR

This paper demonstrates that the entanglement needed to maximally violate Bell inequalities depends on the number of measurement settings, not just outcomes, by establishing new lower bounds on a generalized Grothendieck's constant.

Contribution

It introduces a generalized Grothendieck inequality and proves that entanglement dimension requirements grow with measurement settings, resolving a longstanding conjecture.

Findings

Correlations requiring high entanglement dimension are identified.

Established lower bounds on a new generalized Grothendieck's constant.

Entanglement needed depends on measurement settings, not only outcomes.

Abstract

Suppose Alice and Bob make local two-outcome measurements on a shared entangled state. For any d, we show that there are correlations that can only be reproduced if the local dimension is at least d. This resolves a conjecture of Brunner et al. Phys. Rev. Lett. 100, 210503 (2008) and establishes that the amount of entanglement required to maximally violate a Bell inequality must depend on the number of measurement settings, not just the number of measurement outcomes. We prove this result by establishing the first lower bounds on a new generalization of Grothendieck's constant.